Average Error: 0.1 → 0.1
Time: 14.2s
Precision: 64
\[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
↓
\[\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left({y}^{\frac{1}{3}}\right) \cdot x\right) - y\right) - z\right) + \log t\]
\left(\left(x \cdot \log y - y\right) - z\right) + \log t
↓
\left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left({y}^{\frac{1}{3}}\right) \cdot x\right) - y\right) - z\right) + \log t
double f(double x, double y, double z, double t) {
        double r138841 = x;
        double r138842 = y;
        double r138843 = log(r138842);
        double r138844 = r138841 * r138843;
        double r138845 = r138844 - r138842;
        double r138846 = z;
        double r138847 = r138845 - r138846;
        double r138848 = t;
        double r138849 = log(r138848);
        double r138850 = r138847 + r138849;
        return r138850;
}
↓
double f(double x, double y, double z, double t) {
        double r138851 = x;
        double r138852 = 2.0;
        double r138853 = y;
        double r138854 = cbrt(r138853);
        double r138855 = log(r138854);
        double r138856 = r138852 * r138855;
        double r138857 = 0.3333333333333333;
        double r138858 = pow(r138853, r138857);
        double r138859 = log(r138858);
        double r138860 = r138859 * r138851;
        double r138861 = fma(r138851, r138856, r138860);
        double r138862 = r138861 - r138853;
        double r138863 = z;
        double r138864 = r138862 - r138863;
        double r138865 = t;
        double r138866 = log(r138865);
        double r138867 = r138864 + r138866;
        return r138867;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

    \[\leadsto \left(\left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} - y\right) - z\right) + \log t\]

  4. Applied log-prod0.1

    \[\leadsto \left(\left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

  5. Applied distribute-lft-in0.1

    \[\leadsto \left(\left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} - y\right) - z\right) + \log t\]

  6. Simplified0.1

    \[\leadsto \left(\left(\left(\color{blue}{x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + x \cdot \log \left(\sqrt[3]{y}\right)\right) - y\right) - z\right) + \log t\]

  7. Simplified0.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot x}\right) - y\right) - z\right) + \log t\]
  8. Using strategy rm

  9. Applied pow1/30.1

    \[\leadsto \left(\left(\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \color{blue}{\left({y}^{\frac{1}{3}}\right)} \cdot x\right) - y\right) - z\right) + \log t\]
  10. Using strategy rm

  11. Applied fma-def0.1

    \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left({y}^{\frac{1}{3}}\right) \cdot x\right)} - y\right) - z\right) + \log t\]

  12. Final simplification0.1

    \[\leadsto \left(\left(\mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{y}\right), \log \left({y}^{\frac{1}{3}}\right) \cdot x\right) - y\right) - z\right) + \log t\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))